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One of the most challenging tasks in studying streamflow is quantifying how the complexities of environmental and dynamic parameters contribute to the overall system complexity. To address this, we employed Kolmogorov complexity (KC) metrics, specifically the Kolmogorov complexity spectrum (KC spectrum) and the Kolmogorov complexity plane (KC plane). These measures were applied to monthly streamflow time series averaged across 1879 gauge stations on U.S. rivers over the period 1950–2015. The variables analyzed included streamflow as a complex physical system, along with its key components: temperature, precipitation, and the Lyapunov exponent (LEX), which represents river dynamics. Using these metrics, we calculated normalized KC spectra for each position within the KC plane, visualizing interactive master amplitudes alongside individual amplitudes on overlapping two-dimensional planes. We further computed the relative change in complexities (RCC) of the normalized master and individual components within the KC plane, ranging from 0 to 1 in defined intervals. Based on these results, we analyzed and discussed the complexity patterns of U.S. rivers corresponding to each interval of normalized amplitudes.

One of the most challenging tasks in studying precipitation is quantifying how the complexities of individual components contribute to the overall system complexity. To address this, we employed information measures based on Kolmogorov complexity (KC), specifically the Kolmogorov complexity spectrum (KC spectrum) and the Kolmogorov complexity plane (KC plane). We applied these measures to monthly time series data, both measured and simulated by the EBU POM regional climate model, spanning the period from 1982 to 2005 for Sombor (45.78° N, 19.12° E) in Serbia. The variables analyzed included precipitation—a complex physical system—and its individual components: mean temperature, minimum and maximum temperatures, humidity, wind speed, and global radiation. By applying the listed measures to all time series, we calculated normalized KC spectra for each position in the KC plane, displaying interactive master amplitudes against individual amplitudes. We proposed a simplified four-step method to compute the relative change in complexities within the overlapping area beneath the KC spectra. Our results facilitated a discussion on the relationship between the complexity of precipitation and that of its individual components.

We have used the Kolmogorov complexities and the Kolmogorov complexity spectrum to quantify the randomness degree in river flow time series of seven rivers with different regimes in Bosnia and Herzegovina, representing their different type of courses, for the period 1965–1986. In particular, we have examined: (i) the Neretva, Bosnia and the Drina (mountain and lowland parts), (ii) the Miljacka and the Una (mountain part) and the Vrbas and the Ukrina (lowland part) and then calculated the Kolmogorov complexity (KC) based on the Lempel–Ziv Algorithm (LZA) (lower—KCL and upper—KCU), Kolmogorov complexity spectrum highest value (KCM) and overall Kolmogorov complexity (KCO) values for each time series. The results indicate that the KCL, KCU, KCM and KCO values in seven rivers show some similarities regardless of the amplitude differences in their monthly flow rates. The KCL, KCU and KCM complexities as information measures do not “see” a difference between time series which have different amplitude variations but similar random components. However, it seems that the KCO information measures better takes into account both the amplitude and the place of the components in a time series.

The Kolmogorov complexity of a string is the minimum length of a programme that can produce that string. Information distance between two strings based on Kolmogorov complexity is defined as the minimum length of a programme that can transform either string into the other one, both ways. The second quantised Kolmogorov complexity of a quantum state is the minimum average length of a quantum programme that can reproduce that state. In this paper, a second quantised information distance is defined based on the second quantised Kolmogorov complexity. It is described as the minimum average length of a transformation quantum programme between two quantum states. This distance's basic properties are discussed. A practical analogue of quantum information distance is also developed based on quantum data compression.

When we compress a large amount of data, we face the problem of the time it takes to compress it. Moreover, we cannot predict how effective the compression performance will be. Therefore, we are not able to choose the best algorithm to compress the data to its minimum size. According to the Kolmogorov complexity, the compression performances of the algorithms implemented in the available compression programs in the system differ. Thus, it is impossible to deliberately select the best compression program before we try the compression operation. From this background, this paper proposes a method with a principal component analysis (PCA) and a deep neural network (DNN) to predict the entropy of data to be compressed. The method infers an appropriate compression program in the system for each data block of the input data and achieves a good compression ratio without trying to compress the entire amount of data at once. This paper especially focuses on lossless compression for image data, focusing on the image blocks. Through experimental evaluation, this paper shows the reasonable compression performance when the proposed method is applied rather than when a compression program randomly selected is applied to the entire dataset.

This book has been written in the hope that readers will be able to absorb the key ideas behind algorithmic information theory so that they are in a better position to access the mathematical developments and to apply the ideas to their own areas of interest.

Influenced by stratospheric total ozone column (TOC), cloud cover, aerosols, albedo, and other factors, levels of daily erythemal dose (Her) in a specific geographic region show significant variability in time and space. To investigate the degree of randomness and predictability of Her time series from ground-based observations in Novi Sad, Serbia, during the 2003–2012 time period, we used a set of information measures: Kolmogorov complexity, Kolmogorov complexity spectrum, running Kolmogorov complexity, the largest Lyapunov exponent, Lyapunov time, and Kolmogorov time. The result reveals that fluctuations in daily Her are moderately random and exhibit low levels of chaotic behavior. We found a larger number of occurrences of deviation from the mean in the time series during the years with lower values of Her (2007–2009, 2011–2012), which explains the higher complexity. Our analysis indicated that the time series of daily values of Her show a tendency to increase the randomness when the randomness of cloud cover and TOC increases, which affects the short-term predictability. The prediction horizon of daily Her values in Novi Sad given by the Lyapunov time corrected for randomness by Kolmogorov is between 1.5 and 3.5 days.

Peter Gacs showed (Gacs 1974) that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x. i.e.. C(C(x)|x) ≥ log n — log⁽²⁾ n — O(1). (Here log⁽²⁾ i= log log i.) Following Elena Kalinina (Kalinina 2011). we provide a simple game-based proof of this result: modifying her argument, we get a better (and tight) bound log n — O(1). We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that |x| — C(x) ≤ c and |x| + K(|x|) — K(x) ≥ log ⁽²⁾ |x| — c log ⁽³⁾ |x|. In fact, the results of Solovay and Gacs are closely related. Using the result above, we provide a short roof for Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n — C(x) ≤ c and n + K(n) — K(x) ≥ K(K(n)|n) - 3 K(K(K(n)|n)|n) — c. We also prove a close upper bound K(K(n)|n) + O(1). Finally, we provide a direct game proof for Joseph Miller's generalization (Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set with co-enumerable complement) contains for every length a string of this length, then it contains infinitely many strings x such that |x| + K(|x|) — K(x) ≥ log⁽²⁾ |x| — O(log⁽³⁾ |x|) .

Alluvial channels with sinuosity follow an altered flow behavior, contradictory to straight flows. At the interface of surface water and groundwater, seepage is a significant phenomenon occurring at the boundary of alluvial channels. The study of turbulence in seepage affected sinuous alluvial channels would thus provide us with a better insight into their hydro-morphological behavior. To address the nature of turbulence in sinuous channel with downward seepage an experimental framework was design. The paper reports the structure of turbulence in the sinuous channel for no seepage and seepage flow. With downward seepage, there is a noticeable shift of Reynolds shear stress at near-bed, which reports more momentum transport. The average streamwise and transverse turbulence intensity increased by 3.8–18.5% and 4–10.6%, respectively with downward seepage. Calculation of Kolmogorov complexity and the Kolmogorov complexity spectrum suggests higher randomness in the outer region, which can be associated with excess momentum transport. In the lower flow depth z/h=0.2, the randomness in the transverse velocities is higher in the outer region of the bend for about 25% and 38% compared to the central and inner region of the bend, respectively. With downward seepage, randomness increased especially in the outer region. This increase in randomness may report the erosive action in the outer part of the bend. Permutation entropy provided an informative measure to study the complex behavior of the transverse velocity time-series, which we found to be higher in the outer flow zone. For downward seepage, mean of entropy increased across the bend. The turbulent flow alterations and increase in randomness with seepage may be helpful to understand the flow in seepage affected sinuous alluvial channels.

A positive surge is associated with a sudden change in flow that increases the water depth and modifies flow structure in a channel. Positive surges are frequently observed in artificial channels, rivers, and estuaries. This paper presents the application of Kolmogorov complexity and its spectrum to the velocity data collected during the laboratory investigation of a positive surge. Two types of surges were considered: a undular surge and a breaking surge. For both surges, the Kolmogorov complexity (KC) and Kolmogorov complexity spectrum (KCS) were calculated during the unsteady flow (US) associated with the passage of the surge as well as in the preceding steady-state (SS) flow condition. The results show that, while in SS, the vertical distribution of KC for Vx is dominated by the distance from the bed, with KC being the largest at the bed and the lowest at the free surface; in US only the passage of the undular surge was able to drastically modify such vertical distribution of KC resulting in a lower and constant randomness throughout the water depth. The analysis of KCS revealed that Vy values were peaking at about zero, while the distribution of Vx values was related both to the elevation from the bed and to the surge type. A comparative analysis of KC and normal Reynold stresses revealed that these metrics provided different information about the changes observed in the flow as it moves from a steady-state to an unsteady-state due to the surge passage. Ultimately, this preliminary application of Kolmogorov complexity measures to a positive surge provides some novel findings about such intricate hydrodynamics processes.